Deriving the Beta-Binomial Power Prior Update

Why the Update Needs Care

When a new trial has a small control group, historical control data may provide useful information about the control event probability. A fixed power prior discounts the historical likelihood before combining it with the prior and current-trial likelihood. The resulting beta update is algebraically simple, but its interpretation depends on a substantive assumption: the historical and current data are modeled as observations of the same underlying probability.

This tutorial derives that update for binary outcomes. The discount determines how much the historical likelihood contributes to the posterior parameters. It does not change the observed historical counts, and a discounted contribution need not correspond to a whole number of patients.

Model and Assumptions

Let \(p\) be the probability of the binary event in the control population represented by the current trial. Write \(Y_H=y_H\) for the number of events among \(n_H\) historical control participants, and \(Y_C=y_C\) for the number among \(n_C\) current control participants. The binomial sampling models are

Y_H\mid p \sim \operatorname{Binomial}(n_H,p), \qquad Y_C\mid p \sim \operatorname{Binomial}(n_C,p).

The model treats the two data sets as conditionally independent given \(p\). The baseline prior is \(p\sim\operatorname{Beta}(a,b)\), with \(a>0\) and \(b>0\). A fixed power-prior discount \(\delta\) satisfies \(0\leq\delta\leq1\). The historical likelihood is raised to this power, while the current likelihood is used without discount.

Definition: For a fixed discount \(\delta\), the beta-binomial power-prior update combines the baseline beta prior with the historical binomial likelihood raised to \(\delta\), and then combines that result with the current binomial likelihood. Under the shared-\(p\) model, the posterior distribution is \(\operatorname{Beta}\!\left(a+\delta y_H+y_C,\ b+\delta(n_H-y_H)+(n_C-y_C)\right)\).

This update requires more than binary outcomes. It assumes that the event and observation window are defined comparably, that the relevant control populations can reasonably be represented by one \(p\), and that the historical counts are appropriate for the current estimand. It also assumes a fixed, externally specified \(\delta\); this derivation does not treat the discount as an unknown random parameter.

Deriving the Posterior Parameters

For \(y\) events in \(n\) binomial observations, the likelihood as a function of \(p\) is proportional to \(p^y(1-p)^{n-y}\). The omitted binomial coefficient does not depend on \(p\). Raising the historical likelihood to \(\delta\) therefore gives a kernel proportional to \(p^{\delta y_H}(1-p)^{\delta(n_H-y_H)}\). The current likelihood contributes \(p^{y_C}(1-p)^{n_C-y_C}\), and the beta prior contributes \(p^{a-1}(1-p)^{b-1}\).

\begin{aligned} \pi(p\mid y_H,y_C) &\propto p^{a-1}(1-p)^{b-1} \left[p^{y_H}(1-p)^{n_H-y_H}\right]^\delta p^{y_C}(1-p)^{n_C-y_C}\\ &= p^{a+\delta y_H+y_C-1} (1-p)^{b+\delta(n_H-y_H)+(n_C-y_C)-1}. \end{aligned}

A beta density with parameters \(A\) and \(B\) has kernel \(p^{A-1}(1-p)^{B-1}\) on \(0<p<1\). Matching exponents gives the posterior parameters. They are positive because \(a,b>0\), the counts are nonnegative, and \(\delta\geq0\); thus the resulting beta distribution is proper.

Result: The historical data add \(\delta y_H\) to the beta success parameter and \(\delta(n_H-y_H)\) to the failure parameter. The current data add their full counts, \(y_C\) and \(n_C-y_C\). The posterior mean is \(\left(a+\delta y_H+y_C\right)/\left(a+b+\delta n_H+n_C\right)\).

This is a kernel derivation: factors constant in \(p\) do not affect the posterior shape after normalization. It does not imply that a fractional count represents an actual set of participants. It is a mathematical contribution from the powered likelihood.

Worked Example: A Half-Weighted Historical Control Group

Suppose a hypothetical baseline prior is \(\operatorname{Beta}(1,1)\). In a historical control group, 18 of 30 participants experience the event. The current control group has 12 events among 20 participants. Set the fixed discount to \(\delta=0.5\). Under the stated shared-rate model, calculate the posterior.

The historical group has \(30-18=12\) non-events, and the current group has \(20-12=8\) non-events. The posterior parameters are

\begin{aligned} a_{\text{post}}&=1+0.5(18)+12=22,\\ b_{\text{post}}&=1+0.5(12)+8=15. \end{aligned}

Thus \(p\mid y_H,y_C\sim\operatorname{Beta}(22,15)\). Its posterior mean is \(22/(22+15)=22/37\approx0.5946\). This is the posterior mean event probability under the model, or about 59.46%; it is not a statement that the probability of the event for a particular participant is known exactly.

The historical contribution is 9 event-count units and 6 non-event-count units. Those fractional contributions arise from the discount; the observed historical data remain 18 events and 12 non-events.

Worked Example: Discounting a Larger Historical Group

Consider hypothetical historical data with 8 events among 40 controls and current data with 6 events among 15 controls. Let the baseline prior be \(\operatorname{Beta}(2,8)\) and the fixed discount be \(\delta=0.25\). Check the counts and calculate the posterior mean.

There are \(40-8=32\) historical non-events and \(15-6=9\) current non-events. The update is

\begin{aligned} a_{\text{post}}&=2+0.25(8)+6=10,\\ b_{\text{post}}&=8+0.25(32)+9=25. \end{aligned}

The posterior is \(\operatorname{Beta}(10,25)\), with mean \(10/(10+25)=0.2857\). The historical data contribute 2 event-count units and 8 non-event-count units after discounting. In particular, the 8 in the second line is \(0.25\times32\), not \(0.25\times40\): events and non-events contribute to different beta parameters.

What the Discount Changes

The endpoints clarify the role of \(\delta\). At \(\delta=0\), the historical likelihood contributes no powers of \(p\) or \(1-p\), so the posterior is based on the baseline prior and current data alone. At \(\delta=1\), the historical likelihood is used at full power; under the shared-\(p\) model, historical and current counts both contribute fully. Intermediate values scale each historical count contribution by the same factor.

For example, consider hypothetical data with a \(\operatorname{Beta}(1,1)\) baseline prior, 16 events among 20 historical controls, and 5 events among 20 current controls. With \(\delta=0.5\), there are 4 historical non-events and 15 current non-events, giving

\begin{aligned} a_{\text{post}}&=1+0.5(16)+5=14,\\ b_{\text{post}}&=1+0.5(4)+15=18. \end{aligned}

The posterior is \(\operatorname{Beta}(14,18)\), with mean \(14/32=0.4375\). For comparison, using only the current data gives \(\operatorname{Beta}(6,16)\), with mean \(6/22\approx0.2727\). The difference illustrates why borrowing can matter: when the historical and current event rates differ, the historical data can noticeably shift the posterior. It does not establish that the historical rate is appropriate for the current trial.

Checking the Model and Comparing Alternatives

The update is conjugate under its assumptions, but conjugacy is not evidence that those assumptions hold. Before borrowing, check whether the historical and current control groups have sufficiently comparable populations, outcome definitions, follow-up windows, and event ascertainment. Changes in eligibility criteria, background care, calendar period, or missing-outcome handling may make a shared event probability implausible. Differences in observed rates can flag a concern, although with small groups they may also reflect ordinary sampling variation.

A useful sensitivity check is to recalculate the posterior under several defensible fixed discounts, including \(\delta=0\), and compare the resulting summaries. This reveals how dependent conclusions are on the chosen degree of borrowing. It does not select a discount automatically or repair a mismatch in the estimand or data collection.

Approach	Historical contribution	Key implication
No borrowing	Set \(\delta=0\)	Historical outcomes do not affect the posterior.
Fixed power prior	Use a chosen \(0<\delta<1\)	Historical event and non-event counts are discounted proportionally.
Full pooling	Use \(\delta=1\)	Both groups contribute fully as if they share the modeled event probability.

These are modeling choices, not merely computational settings. No borrowing avoids dependence on historical outcomes but gives up their information. Full pooling may be efficient when a common rate is credible, but can be misleading when the historical controls differ from the current population. A fixed power prior offers a transparent intermediate choice, but its fixed discount does not represent uncertainty about how compatible the data sets are.

Common Mistakes and Limitations

Discounting only historical events. The same \(\delta\) multiplies historical events and historical non-events. The failure parameter must include \(\delta(n_H-y_H)\).
Discounting current data. In this model, only the historical likelihood is raised to \(\delta\). Current counts enter at full weight.
Calling fractional contributions patients. Quantities such as \(\delta y_H\) are algebraic contributions to the posterior parameters, not fractional observed people.
Assuming the formula validates borrowing. The update follows from the specified model; it cannot establish that historical controls represent the current control population.

The method is also limited by the one-rate representation: it does not itself model study-to-study variation, covariate differences, or changes in the event probability over time. If such variation is important, the shared-\(p\) model may be inadequate even when the arithmetic is correct. A fixed discount also leaves the choice of \(\delta\) as a modeling decision that should be justified and examined for sensitivity.

Check Your Understanding

Work through each question, then compare your reasoning with the solutions.

With baseline prior \(\operatorname{Beta}(1,1)\), historical data of 10 events among 25 controls, current data of 4 events among 12 controls, and \(\delta=0.4\), find the posterior parameters.
For the setting in Question 1, explain why the historical failure contribution is not \(0.4\times25\).
What does \(\delta=0\) do to the posterior update, and what does \(\delta=1\) do?
Name two features of a historical control study that should be checked before treating it as informative for the current control event probability.

Solutions. (1) Historical non-events are \(25-10=15\), and current non-events are \(12-4=8\). Thus \(a_{\text{post}}=1+0.4(10)+4=9\) and \(b_{\text{post}}=1+0.4(15)+8=15\), so the posterior is \(\operatorname{Beta}(9,15)\). (2) The failure contribution uses historical non-events, not the total sample size: it is \(0.4(25-10)=6\). (3) At \(\delta=0\), historical data make no contribution; at \(\delta=1\), their counts contribute at full power under the shared-rate model. (4) Examples include comparability of the event definition and follow-up window, similarity of eligibility and patient populations, and consistency of event ascertainment or background care.